The Underlying Dynamics of Credit Correlations
Arthur Berd∗
Robert Engle†
Artem Voronov‡
First draft: June, 2005
This draft: October, 2005
Abstract
We propose a hybrid model of portfolio credit risk where the dynamics of the underlying latent variables is governed by a one factor GARCH
process. The distinctive feature of such processes is that the long-term
aggregate return distributions can substantially deviate from the asymptotic Gaussian limit for very long horizons. We introduce the notion of
correlation spectrum as a convenient tool for comparing portfolio credit
loss generating models and pricing synthetic CDO tranches. Analyzing alternative specifications of the underlying dynamics, we conclude that the
asymmetric models with TARCH volatility specification are the preferred
choice for generating significant and persistent credit correlation skews.
1
Introduction
The credit derivatives market, which exceeds $12 trillion according to most recent estimates from the International Swaps and Derivatives Association [2], encompasses a wide range of instruments, from plain vanilla credit default swaps,
to credit swaptions, portfolio CDS, and synthetic CDO tranches which are becoming a part of the standard toolkit of credit investors [18]. Together with the
growth of the credit derivatives market there has been a great deal of progress
in quantitative modeling for both single-name credit derivatives and for structured credit products (see [14] and [25] for a comprehensive review and further
references).
The latest advances in credit correlation modeling were in part motivated
by the growth and sophistication of the so called correlation trading strategies,
namely strategies involving standardized tranches referencing the Dow Jones
CDX (US) and iTraxx (Europe) broad market indexes. The synthetic CDO
∗ BlueMountain Capital Management, 330 Madison Avenue, New York, NY 10017. E-mail:
aberd@bluemountaincapital.com
† Department of Finance, Stern School of Business, New York University, 44 West Fourth
Street, New York, NY 10012. E-mail: rengle@stern.nyu.edu
‡ Department of Economics, New York University, 269 Mercer Street, New York, NY 10003.
E-mail: artem.voronov@nyu.edu
1
market in general, and the standard tranche market in particular, allows investors to take rather specific views on the shape of the credit loss distribution
of the underlying diversified collateral portfolio. The investor’s views on various slices of this distribution are now well exposed through the pricing of liquid
standard tranches, which in turn is expressed through their implied correlations.
More recently, market participants have switched from using implied correlations defined for each tranche to the notion of base correlation which has
proven useful because it allowed translation of the pricing function of the set
of standard tranches which was a function of two variables (attachment and
detachment points) to a one-dimensional pricing function of the base equity
tranches which only depends on the detachment point. This mapping is similar
to a mapping of the bull spread options with various lower and upper limits onto
a sequence of call options with various strikes — with the base equity tranches
being analogous to a call option on the survival of the portfolio, and the generic
mezzanine tranches being analogous to bull spread options on survival (see [23]
for more details). The base correlation framework has become a de-facto industry standard, and historical comparisons of base correlation levels and its skew
are frequently used to justify investment decisions.
In this paper we propose a simple model of portfolio credit risk with a one
factor GARCH dynamics of loss generating latent variables. Our objectives
in designing the model were to give a plausible explanation to the prominent
correlation skew observed in synthetic CDO markets, and to investigate which
of the properties of the underlying portfolio loss generating models are most
relevant for this task. Our conclusions confirm some of the results known to
analysts in this field, such as the importance of asymmetry in the loss distribution, and provide a substantially more detailed understanding to the origins of
this asymmetry, its dynamics and dependence on term to maturity and other
model parameters.
We begin the paper by providing some motivation for the choice of the model
type in section 2. In particular, we argue that there are many parallels between
modeling of synthetic CDO tranches and modeling of out of money put options
on equity indexes. From these analogies it follows that a dynamic model with
a richer structure than the standard log-normal Black-Scholes-Merton model
must be considered to account for the important features of derivatives traded
in the marketplace, most importantly the volatility skew and term structure
(for equities market) and the correlation skew (for synthetic CDO market). We
conclude that the models of GARCH type have the right properties as candidates
for the underlying dynamics describing credit correlations.
Assuming a factor-GARCH model for single-period returns, we derive analytical formulas for skewness and kurtosis of the cumulative return distributions for a variety of specifications of the single period GARCH process. We
conclude that for sufficiently long horizons (greater than several months) the
effects of stochastic volatility and volatility asymmetry dominate the effects of
non-normality of single-period return shocks. We then demonstrate that the
empirically estimated parameters of the market factor time series, proxied by
the S&P 500 index, do indeed lead to non-Gaussian distribution of cumulative
2
returns for horizons up to 5 or even 10 years.
The connection with credit correlation modeling is made in section 4, where
we show that the pairwise lower tail dependence of equity returns and the pairwise default correlation defined in the latent variable framework via the same
returns are asymptotically equal as the default threshold (tail threshold) is taken
to the lower zero limit. The one factor GARCH model leads to a significantly
different dependence of both measures on the risk threshold compared to the
previously studied copula models. Both the lower tail dependence and the pairwise default correlations are shown to increase at very low thresholds which is
precisely the behavior that would be expected of any model that aims to explain
the steep correlation skew growing toward higher attachment points (i.e. lower
default thresholds).
In section 5 we lay the groundwork for extending our analysis to portfolio
credit risk models by giving a brief introduction to the general copula framework,
the pricing methodology for synthetic CDO tranches, and the large homogeneous
portfolio approximation which we adopt in the rest of the paper. In section 5.2
we argue that the simple pairwise credit correlation is insufficient for description
of the portfolio loss distributions even in the LHP approximation, as it only
relates to the second moment of the distribution, the volatility of losses, and
does not fully specify the shape of the distribution tails. As a tool for a more
complete description of portfolio loss distribution, we introduce the correlation
spectrum measure which both simplifies and extends the widely used notion of
base correlations to a framework suitable for comparison of various default loss
generating models.
In section 6 we compute the correlation spectra for various loss generating
models and use them to study the impact of stylized characteristics of market
factor dynamics on the portfolio credit risk. First, we show that models with
fat tails, such as the static t-copula model, cannot generate upward sloping
correlation skew unless the distribution of the market factor is decoupled from
the distribution of the idisyncratic returns (as it done, for example, in the
double-t copula model), and the latter have thinner tails than the market factor.
Among the dynamic loss generating models, we are able to discriminate between
the specifications of market factor time series and practically rule out those
which do not have an asymmetric volatility process. We demonstrate that the
empirical parameters estimated for S&P 500 as a market factor correspond to
a substantial credit correlation skew in our methodology, thus confirming that
a large portion of the synthetic CDO tranche pricing reflects real risks and not
just risk premia. We conclude the section by examining the dependence of the
correlation skew on various model parameters such as term to maturity and
level of hazard rates — and thereby demonstrate one of the most important
advantages of our methodology, in which the correlation skew is not an input
but an output of the model and therefore its properties and dependencies can
be predicted rather than postulated.
Section 7 presents a brief summary and outline of remaining open questions
and possible extensions of our methodology. The Appendices present additional
proofs and empirical details.
3
2
Modelling Credit and Equity Derivatives
There are many analogies between modeling of equity derivatives and modeling
of credit derivatives in the latent variable framework. While some of these
analogies are superficial, others are intimately related to the structure of the
products and the structure of the models used to price them.
The simplest and often cited analogy is between the implied volatility and
implied correlation. Quoting the implied volatility of an equity option (together
with the level of the underlying stock and the option strike) is equivalent to
quoting its price within the standard Black-Scholes-Merton model. In the same
fashion quoting the implied correlation (together with spread levels of the reference portfolio and the tranche attachment and detachment levels) is equivalent
to quoting the price of a synthetic CDO tranche within the so called Gaussian copula model which has become a de-facto standard in the industry. The
Gaussian copula model as applied to portfolio credit risk was introduced by
Li [19], and extends similar approaches developed earlier for portfolio market
value-at-risk [7], and long-term insurance portfolio loss [11] modeling.
This analogy becomes much deeper and more useful if we focus on the finer
details of derivatives pricing. Just as the observation of a non-trivial implied
volatility surface reflects deviations from the Black-Scholes model assumptions,
the observation of the non-trivial base correlation skew reflects deviations from
the Gaussian copula model assumptions. These assumptions are essentially
equivalent to those of CreditMetrics model of portfolio loss distribution [4]
which, in turn, were derived from an adaptation of Merton’s structural model
of credit risk [22] with corresponding assumption of log-normality of asset returns. In the Gaussian copula model, the multi-variate probability distribution
of times to default is generated as a transformation (with a constant dependence structure) of the multi-variate distribution of asset returns of portfolio
constituents. Thus, it stands to reason that either the assumption of the singlefactor log-normal distribution of asset returns, or the assumption of the constant
dependence structure implied in the Gaussian copula model, or both, are inconsistent with synthetic CDO tranche pricing as reflected by the well established
presence of the base correlation skew.
The observation that using the Gaussian copula model is in principle equivalent to using a version of Merton’s original model is under-appreciated by
many researchers. With this implicit use of Merton’s model also come certain
well-known drawbacks such as the insufficient probabilities of downside risks
for investment grade issuers in the near- and intermediate terms. From the
econometric perspective, the main drawbacks of the classic Merton model are
its inability to account for a number of well established stylized facts regarding
the time series properties of observed equity returns, such as the stochasticity
and persistence of volatilities, asymmetry of volatility response to returns with
levels that are well beyond what that can be explained by the simple capital
structure leverage effect, and the presence of fat tails and other non-Gaussian
features.
The adaptation of the copula-based methodology to reduced-form models
4
80%
19%
70%
17%
15%
50%
imp vol
imp corr
60%
40%
30%
13%
11%
20%
9%
10%
7%
0%
15%-30%
10%-15%
7%-10%
Compound Corr
3%-7%
0%-3%
5%
0%
20%
40%
60%
80%
100%
de lt a
Base Corr
Figure 1: Compounded and base correlation skew for CDX.NA.IG series 3 as of March
2005 (left hand side) and implied volatility skew for S&P 500 index options with 1
year expiry as of March 2005 (right hand side).
of default risk [26], and its re-interpretation in terms of generic latent variable
models [12] have opened the possibility to reconcile the parsimony of the copula
methodology with more flexible models of single-name credit risk. In particular, one no longer has to explicitly assume that the latent variable driving the
generation of default times is log-normally distributed. Among the important
steps towards more realistic modeling of the dependence structure of portfolio
risks within this hybrid framework are the multi-factor Gaussian copula models
[13], the extension to non-Gaussian copulas and in particular to Student-t copulas [20] reflecting the fat-tailed distribution of asset returns, and the explicit
modeling of asymmetric latent variable distributions [1].
A lot of intuition about the shape of the base correlation can be gained by
simply noting that, given a certain level of underlying index spreads, the higher
the attachment point K, the farther out-of-money is the senior tranche (i.e. the
tranche which is exposed to losses above K and up to 1). In the case of the equity
index options the out-of-money put options are typically priced with a higher
level of implied volatility which corresponds to a much fatter downside tails of
the implied return distribution. Similarly, the senior synthetic CDO tranches
are typically priced at a higher level of base correlation which corresponds to
fatter downside tails of default loss distributions (compare the figures in 1, where
we have drawn the correlation skew graph in somewhat unusual way, by placing
the farther out-of-money senior tranches to the left of x-axis to emphasize the
similarity with put options).
Such pricing is commonly attributed to investors’ risk aversion to large loss
scenarios and correspondingly higher risk premia demanded for securities exposing them to such scenarios. However, we believe that it would be unfair to
think of the entire cost premium between various in- and out-of-money tranches
as risk premium and that there are real risks which are being compensated by
these additional costs, albeit perhaps still accompanied by (relatively smaller)
risk premia.
To justify this line of thought let us return for a moment to the case of equity
5
index options and recall that the empirical distribution of returns does indeed
exhibit significant downside tails, and that a large part of the implied volatility
skew can be explained by the properties of the empirical distribution [5]. Let us
list some of the key stylized facts that are known to be relevant for explanation
of the equity index option pricing: 1) the fat tails in the return distribution can
explain the implied volatility smile; 2) the asymmetry in the return distribution
is a necessary ingredient for explaining the implied volatility skew; 3) there
exists an implied volatility surface with non-trivial strike and term structure;
4) the term dependence of the volatility surface is determined by the long-run
aggregated return distribution characteristics which can be significantly different
from those of the short-term (single-period) return distribution; 5) the implied
volatility surface has a much more pronounced skew for stock indexes than for
individual stocks, reflecting a more important role of the common driving factors
compared to idiosyncratic returns in the explanation of the downside risks.
Given the above mentioned analogies between the synthetic CDO tranches
and equity index options, it is quite natural to look for similar stylized facts
that could explain the shape of the base correlation as a function of detachment
level and, potentially, term to maturity, and its key dependencies on the market
and model parameters. As we already noted, the standard Gaussian copula
framework implicitly relies on the Merton-style structural model for definition
of default correlations.
Therefore, if we are to give empirical explanation to the observed base correlation skew we must start by giving an empirical meaning to the variables in
this model. Our working hypothesis in this paper will be that the meaning of
the "market factor" in the factor copula framework is the same as the market
factor used in the equity return modeling. As such, it is often possible to use
an observable broad market index such as S&P 500 as a proxy for the economic
market factor, with an added convenience that there exists a long historical
dataset for its returns and a rich set of equity options data from which one can
glean an independent information about their implied return distribution.
This hypothesis is not uncommon in portfolio credit risk modeling — for
example, the authors of [20] emphasized the importance of using a fat-tailed
distribution of asset returns in the copula framework in part by citing the empirical evidence from equity markets. However most researchers have focused
on the single-period return distribution characteristics.
In contrast, we focus in this paper on the long-run cumulative returns, and
prove that their distribution is quite distinct from that of the short-term (singleperiod) returns. As we will show in the rest of this paper, it is the time aggregation properties and the compounding of the asymmetric volatility responses
that make it possible to explain the credit correlation skew for 5- or even 10year horizons. Moreover, a dynamic explanation of the skew such as presented
in this paper, allows one to make rather specific predictions for the dependence
of this skew on both the term to maturity and on the hazard rates and other
model parameters.
6
3
Time series models of short and long horizon
equity returns
In many applications we first specify time series properties of stock returns for
high frequency time intervals (daily or weekly) and then derive the distribution of stock prices over longer horizons measured in months or even years.
The popular log normal specification that forms the basis of the Black-ScholesMerton option pricing model assumes constant mean returns and volatilities
and iid Gaussian return shocks which leads to the same (log-normal) shape of
the distribution of stock prices for all future horizons.
Models with more realistic dynamics can lead to richer distribution of time
aggregated returns with fat tails and negative skewness even if we assume Gaussian distributions for the return innovations. In particular, models of GARCH
type conform well to the stylized facts regarding both short- and long horizon
equity returns. The autoregressive stochastic volatility process [8] captures the
essence of volatility persistence and clustering observed in the historical time
series. In an extended GARCH framework (see [3] and [9] for a comprehensive review), the non-Gaussian return shocks and the asymmetric response of
volatility to return innovations account for a significant amount of the explanatory power in most model specifications, especially with regard to description of
long-horizon aggregate returns. The term structure of fat-tailness and skewness
of aggregated returns depends on the parametric form chosen for the volatility
process [6].
The volatility dynamics affects not only the marginal distributions of stock
returns but also the distribution of stock co-movements over long horizons or
more generally the copula of long horizon returns. The log-normal model implies
a Gaussian copula for any time horizon whereas multivariate models with more
realistic dynamic properties result in non-Gaussian copulas.
In this paper we focus on two non-Gaussian features of long horizon return
copulas: tail dependence and asymmetry. In this section we describe a simple
one factor model with TARCH(1,1) dynamics that allows us to incorporate
persistence and asymmetry in volatility and correlations and yet is tractable
enough to derive qualitative and quantitative results for non-Gaussian properties
of long horizon return distributions. We begin by describing the univariate
model, and then generalize it to a multi-variate framework with a single factor
structure of returns.
3.1
Univariate model: TARCH(1,1)
Let rt be the log-return of a particular stock or an index such as SP500 from
time t − 1 to time t . zt denotes the information set containing realized values
of all the relevant variables up to time t. We will use the expectation sign with
subscript t to denote the expectation conditional on time t information set:
Et (.) = E (.|zt ) . The time step that we use in the empirical part is 1 day or 1
week. As we already mentioned, predictability of stock returns is negligible over
7
such time horizons and therefore we assume the conditional mean is constant
and equal to zero1 :
mt ≡ Et−1 (rt ) = 0
(1)
The conditional volatility σt2 ≡ Et−1 (rt2 ) of rt in TARCH(1,1) has the autoregressive functional form similar to the standard GARCH(1,1) but with an additional asymmetric term [31]:
rt = σt εt
σt2
=ω+
(2)
2
αrt−1
+
2
αD rt−1
1{rt−1 ≤0}
+
2
βσt−1
We assume that {εt } are iid with zero mean, unit variance, finite skewness
sε and kurtosis kε .We also assume that ω > 0 and α, αD , β are non-negative so
that the conditional variance σt2 is guaranteed to be positive.
Let us introduce the notations for the moments of εt that will be used in
some of the formulas below:
mε ≡ E (εt ) = 0
¡ ¢
vε ≡ E ε2t = 1
¡
¢
vεd ≡ E ε2t 1{εt ≤0}
¡ ¢
sε ≡ E ε3t
¡
¢
sdε ≡ E ε3t 1{εt ≤0}
¡ ¢
kε ≡ E ε4t
¡
¢
kεd ≡ E ε4t 1{εt ≤0}
(3)
The persistence of stochastic volatility in the model is governed by the parameter ζ which is calculated as follows2 :
¡
¢
ζ ≡ E β + αε2t + αD ε2t 1{εt ≤0} = β + α + αD vεd
(4)
If ζ ∈
conditional variance mean-reverts to its unconditional level
¡ [0,¢ 1) then
ω
σ 2 = E σt2 = 1−ζ
. The following parameter ξ will also be useful in describing
the higher moments of TARCH(1,1) returns and volatilities:
¡
¢2
2 d
ξ ≡ E β + αε2t + αD ε2t 1{εt ≤0} = β 2 +α2 kε +αD
kε +2αβ +2αD βvεd +2ααD kεd
(5)
We can rewrite 2 in terms of the increments of the conditional volatility
2
2
∆σt+1
≡ σt+1
− σt2 and the volatility shocks ηt
1 We will discuss later the "risk-neutralization" of the return process which requires certain
drift restrictions in the derivatives pricing context.
2 Note that for ε2 with symmetric distribution v d = 0.5.
ε
t
8
rt = σt εt
¢
¡
= (1 − ζ) σ 2 − σt2 + σt2 ηt
¡
¢
¡
¢
ηt ≡ α ε2t − 1 + αD ε2t 1{εt ≤0} − vεd
(6)
2
∆σt+1
The speed of mean reversion in volatility is 1 − ζ and is small when ζ is
close to one which is usually true for daily and weekly equity returns — hence
the persistence of the stochastic volatility. Using this result, we can estimate
the term dependence of the periodic (short-term) returns variance
¡
¢
2
Et−1 σt+n
= σ2 + ζ n σt2 − σ 2 for n ≥ 0
(7)
The TARCH(1,1) volatility shocks ηt are iid, with zero mean and the following variance:
2
2
var(ηt ) = var(αε2t + αD ε2t 1{εt ≤0} ) = (α + αD ξ) kε + αD
(1 − ξ) ξ (kε + 1) (8)
Persistent and volatile volatility produces fat tails in the unconditional return
distribution even for models with Gaussian shocks. It is easy to see from 6 that
2
conditional volatility of σt+1
is proportional to σt4 and var(ηt )
¡ 2 ¢
¡
¢
vart−1 σt+1
(9)
= vart−1 σt2 ηt = σt4 var(ηt )
The correlation of conditional volatility with the return in the previous period depends on the covariance of return and volatility innovations
¡
¢
αsε + αD sdε
2
corrt−1 rt , σt+1
= corrt−1 (εt , ηt ) = p
var(ηt )
(10)
The negative correlation of return and volatility shocks, often cited as the
"leverage effect"3 , is the main source of the asymmetry in the return distribution. We can see from formula 10 that negative return-volatility correlation
can be achieved either through negative skewness of return innovations sε < 0,
through asymmetry in volatility process αD > 0 or combination of the two. We
call these static and dynamic asymmetry, respectively.
In this paper we are interested in the effects of the volatility dynamics on
the distribution of long horizon returns. While a closed form solution for the
probability density function of TARCH(1,1) aggregated returns is not available,
we can still derive some analytical results for its conditional and unconditional
moments: volatility, skewness and kurtosis.
3 Though
we note here that the magnitudfe of this "leverage effect" in return time series
for stocks of most investment grade issuers far exceeds the amount that would be reasonable
based purely on their capital structure leverage.
9
3.1.1
Volatility
The conditional variance Vt,t+T of the normalized log return Rt,t+T =
√1
T
t+T
X
√1
T
(ln St+T − ln St ) =
ru encompassing T periods from t to t + T follows directly from the
u=t+1
term structure dependence of the periodic return variance 7:
2
Vt,t+T = Et Rt,t+T
⎛
1 ⎝
= Et
T
X
t+1≤u≤t+T
⎞
¡ 2
¢ 1 1 − ζT
σu2 ⎠ = σ 2 + σt+1
− σ2
(11)
T 1−ζ
The unconditional variance is therefore the same as for the short-term returns:
(12)
VT = E(Vt,t+T ) = σ 2
The deviation of the T-horizon conditional volatility Vt,t+T from its unconditional level σ 2 depends on the current deviation of the short horizon volatility
2
σt+1
− σ 2 , aggregation horizon T and the level of volatility persistence ζ. See
figure 2 for illustration.
Conditional Variance Term Structure
2
2
2
σt+1=0.5*σ
2
2
σt+1=σ
2
2
σt+1=2*σ
V t+1,t+T
1.5
1
0.5
0
100
200
300
400
T(w eeks)
500
600
Figure 2: Term structure of conditional variance of time aggregated return Rt+1,t+T .
TARCH(1,1) has persistence coefficient ζ = 0.98 and the followng parametrization:
σ 2 = 1, α = 0.01, αD = 0.10, β = 0.92, εt ∼ N (0, 1). We plotted volatility term
structure for three different initial volatilities: σ 2 /2, σ 2 and 2σ 2
3.1.2
Skewness
Skewness is a convenient measure of return distribution asymmetry. The following proposition gives the formulas for conditional and unconditional third
moments of aggregated returns generated by the TARCH(1,1) model.
10
Proposition 1 Suppose 0 ≤ ζ < 1 and the return innovations have finite skewness, sε , and finite "truncated" third moment, sdε . Then the conditional third
moment of Rt,t+T has the following representation for TARCH(1,1)
T
¡
¢X
1 − ζ T −u ¡ 3 ¢
d
+
α
s
αs
Et σt+u
ε
D ε
3/2
1−ζ
T
T
u=1
u=1
(13)
In addition, if Eσt3 is finite, then unconditional skewness of Rt,t+T is given by
3
Et Rt,t+T
=
1
s
3/2 ε
T
X
¡ 3 ¢
Et σt+u
+
3
∙
¸ ³ ´
3
T
¢
ERt,t+T
1
σt 3
1 ¡
d T (1 − ζ) − 1 + ζ
ST ≡
=
sε + 3 3/2 αsε + αD sε
E
2
2
3/2
1/2
(1 − ζ)
σ
E(Rt,t+T )
T
T
(14)
Proof. See appendix A for the details.
The conditional third moment is a function of the conditional term structure
of σt3 , term horizon T and volatility parameters. The conditional skewness can
be computed using second and third conditional moments derived above. The
asymmetry in the return distribution arises from two sources - skewness of return innovations and asymmetry of the volatility process. Note that the second
term in the formulas for conditional and unconditional skewness is directly related to the correlation of return and volatility innovations. If return-volatility
¡ ¢3
1
correlation is zero (αsε + αD sdε = 0) then ST = T 1/2
sε E σσt . If return innovations are symmetric then asymmetric volatility drives the asymmetry in the
return distribution. In Figure 3 we show conditional and unconditional skewness term structures. For realistic parameters corresponding approximately to
parameters of the TARCH(1,1) estimated for weekly SP500 log returns, both
conditional and unconditional skewness is negative. It decreases in the medium
term, attains the minimum at approximately the 2 year point and then decays
to zero as T increases. The skewness term structure conditional on the high/low
current volatility is above/below the unconditional skewness.
3.1.3
Kurtosis
The fourth conditional moment, if it exists, describes the fat-tailness of the
conditional return distribution and the volatility of return volatility. Formula
9 gives us the conditional volatility of the conditional volatility. The fourth
conditional moment of one period return is proportional to the kurtosis of return
innovations:
4
4
Et rt+1
= σt+1
kε
(15)
For symmetric return shocks and symmetric volatility dynamics (sε = 0
and αD = 0) kurtosis KT has a simple representation in terms of the model
parameters, according to the following proposition.
Proposition 2 If the distribution of εt is symmetric and αD = 0 then uncon-
11
Conditional Skewness Term Structure
0
2
2
σt+1=0.5*σ
2
2
σt+1=σ
-0.2
S
t+1,t+T
2
2
σt+1=2*σ
-0.4
unconditional
-0.6
-0.8
-1
0
100
200
300
T(w eeks)
400
500
600
Figure 3: Term structure of conditional skewness of time aggregated return
Rt+1,t+T .TARCH(1,1) has persistence coefficient ζ = 0.98 and the following
parametrization: σ 2 = 1, α = 0.01, αD = 0.10, β = 0.92, εt ∼ N (0, 1). We
plotted unconditional skewness term structure and conditional for three different ini3
was computed from
tial volatilities: σ 2 /2, σ 2 and 2σ 2 . The term structure of Et σt+u
10,000 independent simulations.
ditional kurtosis of Rt,t+T , if exists, is given by the following formula:
1
γ1 T (1 − ζ) − 1 + ζ T
for T > 1
(K1 − 3) + 6 2
T
T
(1 − ζ)2
1 − ζ2
K1 = kε
1−ξ
KT = 3 +
(16)
where kr and kε are the unconditional kurtosis estimates of one period total
returns rt and return innovations εt , respectively, ζ is the persistence parameter
4, ξ is the parameter defined earlier in 5, and γ1 is the first unconditional autocorrelation coefficient of squared returns, defined in A:
¡ 2
¢
¡
¢
γ1 ≡ corr rt−1
, rt2 = α (kr − 1) + αD krd − vrd + βkr /kε
Proof. See appendix A for the details.
3.2
Multivariate model: One factor ARCH with TARCH(1,1)
factor volatility dynamics
Let us now turn to a multi-variate model of equity returns for M companies,
with a simple dynamic factor structure decomposing the returns into a common
(market) and idiosyncratic components. To concentrate on the time dimension
12
of the model we assume a homogeneity of cross-sectional return properties,
namely that factor loadings and volatilities of idiosyncratic terms are constant
and identical for all stocks. Thus, our homogeneous one factor ARCH model
has the following form.
ri,t = bσm,t εm,t + σεi,t
¡ 2
¢
2
2
= (1 − ρm ) σm
− σm,t
ηm,t
+ σm,t
¡ 2
¢
¡ 2
¢
d
ηm,t ≡ α εm,t − 1 + αD εm,t 1{εm,t ≤0} − vm,ε
(17)
2
∆σm,t+1
where
• b ≥ 0 is the constant market factor loading and it is the same for all stocks
• rm,t is the market factor return with zero conditional mean Et−1 (rm,t ) =
2
2
0, conditional volatility σm,t
≡ Et−1 (rm,t
) and unconditional truncated
¢
¡
d
second moment of return innovations vm,ε ≡ E ε2m,t 1{εm,t ≤0}
• σεi,t are the idiosyncratic return components with constant volatilities σ 2
and zero conditional means Et−1 (σεi,t ) = 0
• {εi,t , εm,t } are unit variance iid for each t and all i
In this model conditional volatilities and pairwise conditional correlations of
stock returns are time varying and depend only on the volatility dynamics of
the market factor.
2
2
2
≡ V art−1 (ri,t
) = b2 σm,t
+ σ2
σi,t
ρ(i,j),t =
Covt−1 (ri,t , rj,t )
q
=
2 σ2
σi,t
j,t
2
b2 σm,t
2
σ2 + b2 σm,t
(18)
(19)
The unconditional correlation between returns of any two stocks is given by
ρ(i,j) =
σ2
2
b2 σm
2
+ b2 σm
(20)
The conditional pairwise correlation ρ(i,j),t is a strictly increasing function
2
of market volatility σm,t
if b > 0 and therefore the persistence and asymme2
try of the market volatility σm,t+1
translates into the persistence and dynamic
asymmetry of the stock correlations ρ(i,j),t .
Because of the simple linear factor structure and constant market loadings
T
X
time aggregated equity returns Ri,T = √1T
ri,u also have a one factor repreu=1
sentation4
Ri,T = bRm,T + Ei,T
(21)
4 To simplify the notations we assume that the initial time t = 0 and use only subscipt for
the time aggregation horizon T.
13
CONDITIONAL CORRELATION TERM STRUCTURE
2
2
σm,0=σm/2
CORRELATION
0.45
2
2
σm,0=σm
0.4
2
2
σm,0=2*σm
0.35
0.3
0.25
0.2
0.15
100
200
300
T(w eeks)
400
500
600
Figure 4: Conditional Correlation Term Structure of time aggregated returns Ri,T
and Rj,T . TARCH(1,1) has persistence coefficient ζm = 0.98 and the following
2
parametrization: σm
= 1, αm = 0.01, αm,D = 0.10, βm = 0.92, εm,t ∼ N (0, 1).
We plotted conditional correlation term structure for three different initial volatilities:
2
2
2
σm
/2, σm
and 2σm
.
where Rm,T =
√1
T
T
X
rm,u and Ei,T =
√1
T
u=1
σ
T
X
εi,u are independent conditional
u=1
on z0 .
The conditional variance, correlation, skewness and kurtosis of aggregated
returns Ri,T can be easily computed in terms of the corresponding moments of
market and idiosyncratic returns.
¡ 2 ¢
Vi,T = E0 Ri,T
(22)
= b2 Vm,T + σ 2
b2 Vm,T
Γ(i,j),T = corr(Ri,T , Rj,T |z0 ) = 2
b Vm,T + σ 2
¡ 3 ¢
¡
¢3/2
E0 Ri,T
3/2
= Γ(i,j),T Sm,T + 1 − Γ(i,j),T
SE,T
Si,T =
3/2
Vi,T
¡
¢ ¡
¢2
Ki,T = Γ2(i,j),T Km,T + 6Γ(i,j),T 1 − Γ(i,j),T + 1 − Γ(i,j),T KE,T
(23)
(24)
(25)
We can see from Figure 4 that indeed the term structure of conditional
pairwise correlation resembles that of the conditional variance in Figure 2.
3.3
SP500 as a proxy for market return
To provide some empirical context to the theoretical discussion above, let us
consider the estimation results of several TARCH(1,1) specifications for SP500
daily and weekly returns. We obtained the daily levels of SP500 from CRSP
14
database. The total number of observations is 10699 and covers the period from
07/02/1962 till 12/31/2004. We constructed daily and weekly log returns and
estimated the parameters of TARCH and GARCH models with Gaussian and
Student-t shocks for 2 samples - full and post-1990.
Tables 1,2,3 in appendix B show estimated parameters and various data
statistics. Note that the Student-t distribution has an additional parameter,
degrees of freedom, that adjusts the tails of the error distribution5 . Since the
Gaussian distribution is nested within the Student-t as a limit of large degrees
of freedom, and since the estimates of the full unconstrained model result in a
relatively small and statistically significant value of the degrees of freedom, we
conclude that the data points toward the fat-tailed return shock distribution.
On the other hand, the asymmetric TARCH model is nested within the
symmetric GARCH in the limiting case αD = 0. The estimated asymmetric
coefficient αD in the TARCH model is not only non-zero, but significantly higher
than the symmetric coefficient α for both complete and post 1990 samples, both
daily and weekly frequencies and Gaussian and Student-t shock distributions.
Thus, we conclude that the asymmetric volatility is prominently present in the
data. The best fit model among those considered is the TARCH(1,1) with
Student-t distribution of return innovations. The additional parameters of this
model are statistically significant.
In Figure 5 we show the estimate of skewness for overlapping returns of
different aggregation horizons measured in days. The full sample shows high
negative skewness for one day return because of the 1987 crash. On the post 1990
sample negative skewness rises with aggregation horizon up to 1 year and then
slowly decays toward zero. Both samples show significant skewness for horizons
of several years. We should note that confidence bounds around skewness curves
are quite wide due to the persistence and high volatility of the squared returns
and serial correlation of the overlapping observations.
To make sure that asymmetry in volatility is not a result of several extreme
negative returns like 1987 crash we provide data statistics and re-estimated
parameters of TARCH models for trimmed full and post 1990 samples. The
trimming is done by cutting excess volatility in the most extreme 0.05% observations of both positive and negative return. We can see from Tables 1b and 2b
that trimming significantly reduced skewness sr of daily returns but the volatility of daily returns is still significantly asymmetric. Weekly returns are less
affected by trimming both in terms of TARCH parameters and unconditional
skewness and kurtosis. However, the long-run skewness of aggregate returns
remains largely unaffected by the trimming because it is driven mostly by the
asymmetry of the volatility and the value of αD does not change much due to
trimming, especially for the model versions with Student-t distributed return
shocks.
5 The Student-T distribution is sometimes critiqued as a model for continously componded
returns because the expectation of the exponent of Student-T variable is infinite and therefore
expected return over one period is also infinite. In practice (estimation) we can think of
Studen-T distribution as being truncated at far enough tails so that the estimation procedure
is not changed, while the expectation of the exponent is finite.
15
0
-0.2
-0.4
S
T
-0.6
-0.8
-1
1962-2004
1990-2004
-1.2
-1.4
0
100
200
300
400
T(days)
500
600
700
800
Figure 5: Term structure of skewness for SP500 time aggregated log returns estimated
with overlapping samples moments for full and post-1990 data.
4
Modeling Tail Risk and Default Correlation
The dynamic models of aggregate equity returns presented in the previous section can serve as an important ingredient for modeling of tail risks and default
correlations. In this paper we are interested in the effects of the return dynamics
on the joint distribution of RT = [R1,T , ..., RK,T ]06 . Denote
• FT (di ) ≡ P (Ri,T ≤ di |z0 ) conditional cdf of aggregate total returns Ri,T
• GT (di ) ≡ P (Ei,T ≤ di |z0 ) conditional cdf of aggregate idiosyncratic returns Ei,T
• FT (d) ≡ P (RT ≤ d|z0 ) joint conditional cdf of RT
¡
¢
• CT (u) ≡ FT FT−1 (u1 ) , ..., FT−1 (uM ) conditional copula of RT
Note than the assumption of one factor structure implies that equity returns
RT are independent conditional on the market return Rm,T and therefore FT (d)
can be computed as expectation of the product of conditional cdfs e.g. for
unconditional7 distribution:
ÃM
!
ÃM
!
_
Y_
Y_
F T (d) = E
P (Ri,T ≤ di |Rm,T ) = E
GT (di − bi Rm,T )
(26)
i=1
i=1
The tail dependence coefficient and the "default correlation" coefficient are
convenient measures of the risk of joint extreme movements for a pair of assets.
bold letters denote N dimentional vectors e.g. x≡ [x1 , ..., xN ]0 .
unconditional distributions
and copulas we use the same notations but with bar above
_
the corresonding letter, e.g F T .
6 the
7 for
16
0.2
ρ
D
0.15
0.1
TARCH
Gaussian
0.05
GARCH
T-Copula (v =12)
0
0
0.05
0.1
0.15
0.2
p
Figure 6: Default correlation as a function of p for TARCH, GARCH,TCopula and Gaussian models are calculated using 100,000 Monte Carlo simulations.
The linear correlation parameter is 0.3 for all 4 models.
TCopula degrees of freedom parameter is 12.
TARCH and GARCH market factors correspond to 5 year log returns which are computed based
on the returns simulated over weekly intervals. TARCH(GARCH)parameters
are α = 0.01(0.06), αD = 0.1(0), β = 0.92(0.92), Gaussian shocks and idiosyncrasies.
Suppose Ri,T and Rj,T are the stock returns for companies i and j over the [0, T ]
time horizon. The coefficient of lower tail dependence and the default correlation
coefficient for two random variables with the same continuous marginal cdfs,
FT (R) , are defined as
CT (p, p)
p
C
(p,
p)
−
p2
T
ρD
i,j (p) = corr(1{Ri,T ≤dp } , 1{Rj,T ≤dp } ) =
(1 − p)p
λD
i,j = lim P (Ri,T ≤ dp |Rj,T ≤ dp ) = lim
p→+0
p→+0
(27)
(28)
where p is the probability of crossing the threshold (also interpreted as the default probability), and is related to the latter via the relationship dp = FT−1 (p) .
Both measures depend only on the bivariate copula of the two random variables
D
and are asymptotically equal: lim ρD
i,j (p) = λi,j .
p→+0
On Figure 6 we show the default correlation ρD
1,2 as a function of p for 4
different models - TARCH, GARCH, Gaussian and Studen-t copula. For all
4 models the linear correlation of latent returns is set to 0.3. The Gaussian
copula is symmetric and have zero tail dependence for both upper and lower
tails. We can see on the graph that it also has lowest default correlation for
all default probabilities in the range of [0,0.2]. The Student-t copula is also
symmetric but has fatter joint tails compared to the Gaussian copula. Its default
correlation is above the Gaussian for all p and converges to a positive number
17
(the tail dependence coefficient) as p decreases to zero. TARCH and GARCH are
calibrated to have volatility dynamics parameters corresponding approximately
to the weekly SP500 returns and the time aggregation horizon is set 5 years.
We can see that TARCH has higher default correlation than other 3 models
and for very low quantiles is upward sloping. The upturn for the extreme
tails is a consequence of the left tail shape of the common factor. The default
correlation for very low default probabilities should be close to 1 since the left
tail of the factor is fatter than the left tail of the idiosyncratic shocks. The
GARCH default correlations are closer to the Gaussian because the 5 year time
aggregated market factor is "almost" Gaussian. As we showed in the previous
sections both kurtosis and skewness of the market factor converge to zero faster
for GARCH than TARCH given the same level of volatility persistence.
On Figure 7 we show default correlation ρD
1,2 as a function of p for the
TARCH model but for different aggregation horizons. The default correlations
for 1 and 5 year horizons are significantly above the 1 week horizon. The term
structure of skewness for this example is shown on Figure 3 in the previous
section. We can see from the skewness term structure figures that weekly returns are symmetric whereas time aggregated returns for longer horizons(1 and
5 years) have significant negative skewness which increases the default correlations.
0.2
0.18
ρD(p)
0.16
0.14
0.12
T=1week
T=1 y ear
0.1
T=5 y ears
0.08
0
0.05
0.1
p
0.15
0.2
Figure 7: Default correlation as a function of p for the TARCH model for 1 week,
1 year and 5 year time horizons which are calculated based on 100,000 Monte Carlo
simulations of weekly returns. The linear correlation is 0.3. TARCH parameters are
α = 0.01, αD = 0.1, β = 0.92, Gaussian shocks and idiosyncrasies.
18
5
5.1
Modeling Portfolio Credit Risk
General Copula Framework
In this section we describe a hybrid semi-dynamic approach to modeling of default correlations in a large homogeneous portfolio of credit exposures. Consider
a portfolio of M credit-risky obligors. We start with a static setup with a fixed
time horizon [0, T ] and to simplify notation skip the time subscript for time
dependent variables. At time t = 0 all M obligors are assumed to be in nondefault state and at time T firm i is in default with probability pi . We assume we
0
know the individual default probabilities p = [p1 , ..., pM ] (either risk-neutral,
e.g. inferred from default swap quotes, or actual, e.g. estimated by rating agencies). Let τi ≥ 0 be the random default time of obligor i and Yi = 1{τi ≤T } the
default dummy variable which is equal to 1 if default happened before T and 0
otherwise.
The loss generated by obligor i conditional on its default is denoted as li > 0.
The loss li is a product_ of the total
exposure size ni and percentage losses in
_
case default occurs 1 − Ri where Ri ∈ [0, 1] is the recovery rate. We also assume
that li is constant (see [1] for discussion on stochastic recoveries). Portfolio loss
LM at time T is the sum of the individual losses for the defaulted obligors
LM =
M
X
i=1
li 1{τi ≤T } =
M
X
li Yi
(29)
i=1
The mean loss of the portfolio can be easily calculated in terms of individual
default probabilities:
E (LM ) =
M
X
li E (Yi ) =
i=1
M
X
l i pi
(30)
i=1
Risk management and pricing of derivatives contingent on the loss of the
credit portfolio, such as CDO tranches, require knowing not only the mean
but the whole distribution of portfolio loss with cdf FL (x) = P (LM ≤ x) .
Portfolio loss distribution depends on the joint distribution of default indicators
0
Y = [Y1 , ..., YM ] and in a static setup can be conveniently modeled using the
latent variables approach [12]. Particularly, to impose structure on the joint
distribution of default indicators we assume that there exists a vector of M
0
real-valued random variables R = [R1 , ..., RM ] and M dimensional vector of
0
non-random default thresholds d = [d1 , ..., dM ] such that
Yi = 1 ⇐⇒ Ri ≤ di for i = 1, ..., M
(31)
Denote F : RM → [0, 1] as a cdf of R and assume that it is a continuous
function with marginal cdf {Fi }M
i=1 . For each obligor i the default threshold di
is calibrated to match the obligor’s default probability pi by inverting the cdf of
its aggregate returns Ri : di = Fi−1 (pi ). According to Sklar’s theorem, under
19
the continuity assumption F can be uniquely decomposed into marginal cdfs
M
{Fi }M
⇒ [0, 1]
i=1 and the M -dimensional copula C : [0, 1]
F (d) = C (F1 (pi ) , .., FM (pM ))
(32)
Several popular copula choices are the Gaussian copula model [19], Student-t
[20] and Clayton:
• Gaussian copula
¢
¡
C G (p; Σ) = ΦΣ Φ−1 (p1 ) , ..., Φ−1 (pM )
• Student-t copula
−1
C T (p; Σ, v) = tΣ,v (t−1
v (p1 ) , ..., tv (pM ))
• Clayton
Ã
M
X
C Cl (p) = max 1 − M +
p−β
i
i=1
!β
The choice of copula C defines the joint distribution of default indicators
from which the portfolio loss distribution can be calculated. The number of
names in the portfolio can be large and therefore the calibration of the copula
parameters can be problematic. To reduce the number of parameters some form
of symmetry is usually imposed on the distribution of default indicators. Gordy
[15] and Frey and McNeil [12] discuss the mathematics behind the modeling of
credit risk in homogeneous groups of obligors and equivalence of the homogeneity
assumption to the factor structure of default generating variables. Conditional
on the factors, defaults are independent and the conditional joint distribution of
default indicators can be easily calculated using the multinomial distribution.
To simplify the calculations even more, a large homogenous portfolio (LHP)
approximation can be used to approximate the multinomial distribution with a
finite number of obligors. Schonbucher and Shubert [26] and Vasicek [29] show
that LHP approximation is quite accurate for upper tail of the loss distribution
even for mid-sized portfolios of about 100 names. We use symmetric one factor
LHP setup in this paper for analytical tractability.
Assumption_ 1(Symmetric One Factor Model): Assume that loss given
default li = (1− Ri )ni and individual default probabilities pi are the same for all
M names in the portfolio and that the latent variables admit symmetric linear
one factor representation:
ni = n
_
(33a)
_
Ri = R
pi = p
Ri = bRm +
p
1 − b2 Ei with 0 ≤ b ≤ 1
20
(33b)
(33c)
(33d)
where Rm and Ei are independent zero mean, unit variance random variables.
E 0i s are identically distributed with cdf G(.).
Parameter b defines the pairwise correlation of latent variables which is often
referred to as "asset correlation" (this naming reflects the interpretation of latent
variables as asset returns in Merton-style structural default models):
ρ = b2
(34a)
Suppose we increase the number of names in the portfolio while keeping
the total exposure size of the portfolio constant so that ni = N/M . Conditional on Rm the loss of the portfolio contains
mean of independent iden³ the
_´
PM
1
tically distributed random variables, LM = 1 − R N M
i=1 1{Ri ≤d} , which
a.s. converges to its conditional expectation as M increases to infinity. We use
L without subscript to denote the portfolio loss under LHP assumption.
Proposition 3 (LHP Loss) Under Assumption 1
"
#
M
³
³
_´
_´
1 X
L ≡ lim
1{Ri ≤d} = 1 − R N P (Ri ≤ d|Rm )
1−R N
M →∞
M i=1
µ
¶
³
_´
d − bRm
a.s. for any Rm ∈ supp (G)
= 1 − R NG √
1 − b2
(35)
Proof. see proposition 4.5 in [12]
Based on 35 cdf of L can be expressed in terms of the cdf of Rm
P (L ≤ l) = P (Rm ≥ d1 (l))
⎛
⎞
√
l
d
1 − b2 −1 ⎝
⎠
³
d1 (l) = −
G
_´
b
b
1−R N
(36)
(37)
The probability that a diversified portfolio will incur a small loss is high when
the probability of market return Rm falling below barrier d1 is low. In other
words a small loss corresponds to the right tail of the market return distribution.
The left tail of the market return distribution corresponds to large portfolio
losses — the thicker the left tail the more probable is a large loss. The market
factor threshold d1 depends on the single name default barrier d, market factor
loadings b and the loss-per-obligor parameters. Note that d1³is not
necessarily
_´
monotonic function of b. Only for small losses, such that l < 1 − R N G(0), it
is increasing in b.
For Gaussian copula we have familiar formula for the LHP loss derived by
Vasicek [29], where we have substituted the asset correlation parameter in place
21
of the factor loading using the relation 34a:
µ −1
¶
√
³
_´
Φ (p) − ρRm
G
√
L = 1 − R NΦ
1−ρ
¡ G ¢
P (L ≤ l) = 1 − Φ d1 (l)
⎛
⎞
√
−1
l
Φ (p)
1−ρ
⎠
dG
− √ Φ−1 ⎝ ³
_´
√
1 (l) =
ρ
ρ
1−R N
5.2
(38)
(39)
(40)
From Loss Distributions to Correlation Spectrum
The mean of the loss distribution is not affected by the choice of copula. The
second and higher moments of the loss distribution depend on the copula characteristics. In particular, the variance of the loss can be expressed in terms of
bivariate default correlation coefficients, ρD (p) , defined in Section 4.
!
Ã
M
_
1 X
(41)
1{Ri ≤dp }
V ar(LM ) = V ar (1 − R)N
M i=1
µ
¶
_
1
M −1 D
= (1 − R)2 N 2 p(1 − p)
+
ρ (p)
M
M
_
V ar(L) = (1 − R)2 N 2 p(1 − p)ρD (p)
(42)
where ρD (p) = corr(1{Ri ≤dp } , 1{Rj ≤dp } ). By comparing the default correlation
coefficients, as we did in Section 4, we therefore implicitly compare the impact
of the copula choice on the loss variance.
In addition to the variance of L we are also interested in measuring(pricing)
the extreme risks - the likelihood of small and large losses. To do that we define
the loss tranches which allow us to look at the particular slices of portfolio loss.
Let (Kd , Ku ] denote a tranche with attachment point Kd and detachment point
Ku expressed as fractions of the reference portfolio notional so that 0 ≤ Kd <
Ku ≤ 1. The notional of the tranche, N(Kd ,Ku ] , is N (Ku − Kd ) where N is the
notional of the portfolio. The loss L(Kd ,Ku ] of the tranche is the fraction of L
that falls between Kd and Ku . For simplicity assume that total notional N is
normalized to 1.
L(Kd ,Ku ] = f(Kd ,Ku ] (L)
f(Kd ,Ku ] (x) ≡ (x − Kd )+ − (x − Ku )+
(43)
(44)
Tranches with zero attachment point, (0, Ku ] , and unit detachment point,
(Kd , 1] , are called equity and senior tranches correspondingly. Loss of any
tranche can be decomposed into losses of two equity tranches L(Kd ,Ku ] = L(0,Ku ] −
L(0,Kd ] . Expected loss of the equity tranche (0, K] depends on the portfolio loss
distribution and in LHP approximation can be computed using only the distribution of the market factor
22
¶
¸
∙ µ
³
_´
d − bRm
1{Rm ≥d1 (K)} +KP (Rm < d1 (K))
EL(0,K] = Ef(0,K] (L) = 1 − R E G √
1 − b2
(45)
The expectation in (45) can be computed by Monte Carlo simulation or numerical integration if we know G and distribution of Rm (see appendix C). For the
Gaussian copula, the integral can be calculated in a closed form.
³
_´ ¡
√ ¢
E G L(0,K] = 1 − R Φ Φ−1 (p) , −d1 ; − ρ + KΦ (d1 )
(46)
√
µ
¶
1−ρ
1
K
_
(47)
d1 = √ Φ−1 (p) − √ Φ−1
ρ
ρ
1−R
Because of its analytical tractability, it is convenient to use the Gaussian
copula as a benchmark model when comparing different choices of dependence
structure. The asset correlation parameter ρ plays a similar role to the implied
volatility for equity options because an equity tranche (0, K] is essentially a
call option on the surviving part of the underlying portfolio. Like any long
call option position, its value is increasing as a function of the uncertainty of
the underlying. The underlying in this case is the distribution of the portfolio
losses, and its uncertainty increases with the asset correlation parameter, as can
be seen from:
³
³
_ ´2
_ ´2 £ ¡
¢
¤
V ar(L) = EL2 − 1 − R p2 = 1 − R
Φ Φ−1 (p) , Φ−1 (p) ; ρ − p2
Note also that a similar dependence on bivariate default correlation ρD was
derived in 42. Thus, by finding the asset correlation level that in certain sense
replicates the results of more complex portfolio loss generating models in the
context of a Gaussian copula framework, we can translate the salient features
of such models into mutually comparable units. More specifically, we define the
correlation spectrum as follows:
Definition 4 Suppose
distribution of a large homogeneous portfolio is
n the loss
_o
generated by a model C, p, R with copula C, equal individual default probabilih
³
³³
_
_´
_ ´ ´i
ties p and recovery rate R. Let L(0,K] ∈ pf(0,K] 1 − R , f(0,K] 1 − R p
be the expected loss
of the equity ntranche
(0, K] . We define the correlation
_
_o
spectrum ρ(K, p, R) of the model C, p, R as the correlation parameter of the
Gaussian copula that produces the same expected loss EL(0,K] for the tranche
(0, K] for the given horizon T and given single-issuer default probability p
_
ρ(K, p, R) solves E G L(0,K] (ρ) = EL(0,K] for all K ∈ [0, 1]
where EL(0,K] is expected loss of the tranche
n
_o
(0, K] generated by model C, p, R
where E G L(0,K] is defined in (46).
23
(48)
The correlation spectrum as defined above is closely related but not identical
to the notion of the base correlation used by many practitioners [23]. The
difference is that the base correlation is defined using the prices of the equity
tranches, which in turn depend on interest rates, term structure of losses, etc.
By contrast, the correlation spectrum is defined without a reference to any
market price. It characterizes the portfolio loss generating model, rather than
the the supply/demand forces in the market. Therefore we believe it is a more
convenient tool for comparing different models, while the base correlation is
presumably better for comparing the relative value between actual tranches.
Another important point is that the correlation spectrum depends implicitly
on the term to maturity via the cumulative default probability p. However, this
is not the only dependence — potentially, the dependence structure characterized
by the copula C also exhibits some time dependence when viewed within the
context of the Gaussian copula. This statement needs a clarification — the copula
C itself is defined in a manner that encompasses all time horizons and therefore
cannot depend on any particular horizon. However, when we translate the
tranche loss generated with this dependence structure into the simpler Gaussian
model the transformation that is required may depend on the horizon. As we will
see in subsequent sections, this is indeed the case for the portfolio loss generating
models based on GARCH dynamics, which are therefore characterized by a nontrivial term structure of the correlation spectrum.
To ensure that the correlation spectrum is well defined we need to prove that
(48) has a unique solution. Let us first prove the following:
Proposition 5 For the Gaussian copula, the expected loss of an equity tranche
is a monotonically decreasing function of ρ and attains its maximum (minimum)
when correlation is equal to 0 (1).
24
Proof. Using (46) and the properties of Gaussian distribution8 we derive
∂ G
(49)
E L(0,K]
∂ρ
∙
¸
³
_´
¡ −1
¡ −1
1
√ ¢
√ ¢ ∂
=− 1−R
d1
√ Φ3 Φ (p) , −d1 ; − ρ + Φ2 Φ (p) , −d1 ; − ρ
2 ρ
∂b
∂
+ Kφ (d1 ) d1
∂b
_
1 − R ¡ −1
√ ¢
= − √ φ Φ (p) , −d1 ; − ρ < 0
2 ρ
EρG L(0,K] ≡
for any ρ ∈ (0, 1).
Therefore, there is a one-to-one mapping between loss distribution and correlation spectrum, and our transformation does not lead to any loss of information.
The next proposition shows how to calculate the loss cdf using the correlation
spectrum and its slope along the K-dimention.
n
_
_o
Proposition 6 Suppose ρ(K, p, R) is the correlation spectrum for model C, p, R
and the probability distribution function of the portfolio loss is a continuous
function then the loss cdf can be computed from the correlation spectrum:
_
P (L ≤ K) = P G (L ≤ K) + ρK (K, p, R)EρG L(0,K]
(50)
P G (L ≤ K) = 1 − Φ (d1 )
³
_´ 1
¡
√ ¢
EρG L(0,K] = 1 − R √ φ Φ−1 (p) , −d1 ; − ρ
2 ρ
√
µ
¶
K
1−ρ
1
_
d1 = √ Φ−1 (p) − √ Φ−1
ρ
ρ
1−R
(51)
where
(52)
(53)
_
and the functions are evaluated at ρ = ρ(K, p, R).
8 The
following properties of 2 dimentional Gaussian cdf are used in the calculation
!
Ã
x − ρy
Φ2 (x, y; ρ) = φ (y) Φ p
1 − ρ2
∂
Φ (x, y; ρ) = φ (x, y; ρ)
∂ρ
where φ (.) denotes, depending on the number of agruments, pdf of standard Normal distribution and pdf of bivariate Normal with standard Normal marginals and correlation coefficient
as a third argument. Numerical subscript denotes the partial derivative with respect to the
corresponding argument. First formula is straitforward. The proof of the second can be found
in Vasicek([30])
25
Proof. first note that the derivative with respect to K of the expected tranche’s
loss under true copula C is related to the cdf of the loss
¢
¡
d
d
EL(0,K] =
E L − (L − K)+
dK
dK
d
=−
E (L − K)+ = E1{L−K≥0} = 1 − P (L ≤ K)
dK
(54)
therefore
d
EL(0,K]
dK
_
G
L(0,K] − ρK (K, p, R)EρG L(0,K]
= 1 − EK
P (L ≤ K) = 1 −
(55)
_
_
1−R ¡
√ ¢
= 1 − Φ (d1 ) + √ φ Φ−1 (p) , −d1 ; − ρ ρK (K, p, R)
2 ρ
³
_´ 1
_
¡
√ ¢
= P G (L ≤ K) + 1 − R √ φ Φ−1 (p) , −d1 ; − ρ ρK (K, p, R)
2 ρ
where partial derivative with respect to K is computed as
∂ G
(56)
E L(0,K]
∂K
³
´
_
¡
∂
√ ¢ ∂
= − 1 − R Φ2 Φ−1 (p) , −d1 ; − ρ
d1 + Kφ (d1 )
d1 + Φ (d1 )
∂K
∂K
= Φ (d1 )
G
EK
L(0,K] ≡
We defined the correlation spectrum for the loss distribution with a fixed
time horizon. In the next section we illustrate the pricing of portfolio tranche
swap contracts and show that the value of the swap depends on the whole term
structure of the correlation spectrum up to the maturity of the swap.
5.3
Pricing of Synthetic CDO Tranches
In this section we briefly define the payoff structure of synthetic CDO tranche
contracts and their pricing. Consider a synthetic CDO with fixed maturity T
written on a synthetic portfolio. The loss L(Kd ,Ku ] (t) of the tranche (Kd , Ku ]
at time t ≤ T is a fraction of portfolio loss L (t) that falls between Kd and Ku .
L(Kd ,Ku ] (t) = f(Kd ,Ku ] (L(t))
(57)
The swap contract for a particular tranche is swap of cash flows between the
"premium leg" and the "protection leg". The protection buyer agrees to pay a
fixed fee s to the protection seller in the proportion to the survived notional of
the tranche. The protection seller compensates the tranche losses to the insured
until the maturity of the contract.
26
Since the swap contract is a contingent claim on the portfolio loss it can be
priced using the risk-neutral distribution of the portfolio losses. We assume that
interest rate risk is not correlated with credit risk and denote by D (0, t) the
price at time 0 of a zero coupon bond maturing at time t. The payoff structure
of both premium and protection legs is linear in the tranche’s loss and therefore
to price these legs when the interest rates are not correlated with default risk we
only need to know the term structure of expected tranche losses. Introduce the
tranche’s default probability P(Kd ,Ku ] (t) as expected fraction of the tranche’s
notional that is lost due to defaults by time t.
¤
£
E0 L(Kd ,Ku ] (t)
P(Kd ,Ku ] (t) =
(58)
N(Kd ,Ku ]
For simplicity assume that time is continuous. The value of the protection
leg at time 0
!
ÃZ
V0protection = E0
T
D (0, t) dL(Kd ,Ku ] (t)
(59)
t=0
= N(Kd ,Ku ]
Z
T
D (0, t) dP(Kd ,Ku ] (t)
t=0
Assuming that protection fee is paid in ∆q intervals e.g. quarterly the value
of the premium leg at time 0
⎞
⎛
T /∆q
X
£
¤
D (0, q∆q) N(Kd ,Ku ] − L(Kd ,Ku ] (q∆q) s∆q ⎠ (60)
V0premium = E0 ⎝
q=1
T /∆q
= N(Kd ,Ku ] s∆q
X
q=1
£
¤
D (0, q∆q) 1 − P(Kd ,Ku ] (q∆q)
The par spread of the swap contract is the spread that makes the values of
protection and premium legs equal. As we already mentioned swap contract
cash flows are linear functions of the tranche losses and therefore the values of
the both legs depend only on the tranche’s expected losses. Because timing of
the losses is important when the interest rates are not zero we need the whole
term structure of the tranche’s expected losses up to the maturity of the swap
to price the contract.
The portfolio loss L(t) at time t in the Gaussian copula framework depends
on time t only through the single-issuer default probability pt 9 . As we showed
in proposition 2 the correlation spectrum is the equivalent representation of the
loss distribution for
a fixed time horizon. The dependence of the correlation
_
spectrum ρ(K, p, R) on t is also achieved through the second argument - singleissuer default probability p. For example the expected tranche loss at time t
9 The portfolio loss generating copula does not change with T because it corresponds to
the copula of time to default distribution which by definition does not depend on T.
27
is³the expected
tranche loss of the Gaussian model with correlation parameter
_´
ρ K, pt , R and single-issuer default probability pt :
³ ³
´
_´
EL(0,K] (t) = E G L(0,K] ρ K, pt , R , pt
(61)
The expected tranche loss that happens between t and t + dt can therefore
be computed from the correlation spectrum using its level and the slope in the
p-dimention:
dE G L(0,K] (t)
dpt
dpt
´
³
³
_´
= EpG L(0,K] (t) + ρp K, pt , R EρG L(0,K] (t) dpt
dEL(0,K] (t) =
where
(62)
_
1−R ¡
√ ¢
EρG L(0,K] (t) = − √ φ Φ−1 (pt ) , −d1 ; − ρ
2 ρ
µ
¶
√
³
_´
−d1 + ρΦ−1 (pt )
∂ G
G
√
Ep L(0,K] (t) ≡
E L(0,K] (t) = 1 − R Φ
∂p
1−ρ
µ
¶
√
K
1 − ρ −1
1 −1
_
d1 = √ Φ (pt ) − √ Φ
ρ
ρ
1−R
(63)
(64)
(65)
_
and the functions are evaluated at ρ = ρ(K, p, R). Thus, both legs of the
swap contract can be priced using the correlation spectrum surface, single-issuer
default probability term structure and the formulas (58) -( 62).
6
Comparing Portfolio Loss Generating Models
In section 4 we demonstrated that dynamic models such as GARCH and TARCH
can produce significant pairwise default correlation even for very low default
thresholds. Thus, one can hope that these models should also be able to capture the important aspects of multi-variate default losses in a diversified portfolio
setting. But in order to discriminate between these models and to understand
which of their characteristics are the most important from a credit modeling
perspective, one must have a good measure that makes such comparisons not
only possible but hopefully apparent and intuitive. The market standard measure is the base correlation. However, this measure is best suited for comparison
of pricing of similar tranches rather than comparison of different models. In particular, base correlation implicitly depends on the level and term structure of
interest rates, as well as conventions such as coupon payment frequency, up-front
pricing, etc.
Our goal in this paper is not so much to price a specific set of tranches
under given market conditions as to provide a general framework for judging
28
the versatility of various dynamic portfolio credit risk models. All such models,
whether defined via dynamic multivariate returns model like in this paper or in
various versions of the static copula framework ([19], [12], [26], [20], [13], [1]), can
be characterized by the full term structure of loss distributions. Thus, without
loss of generality, we can refer to all models of credit risk as loss generating
models, with an implicit assumption that any two models that produce identical
loss distributions for all terms to maturity are considered to be equivalent. The
correlation spectrum, introduced in section 5.2, conveniently transforms specific
choice of a loss generating model into a two dimensional surface ρ(K, T ) of the
Gaussian copula correlation parameter, with the main dimensions being the loss
threshold (detachment level) K and the term to maturity T . All other inputs
such as the recovery rate R, the term structure of (static) hazard rates h, the
level of linear asset correlation ζ, the Student-t degrees of freedom ν, various
GARCH model coefficients, etc. — are considered as model parameters upon
which the two-dimensional correlation spectrum itself depends. Note that in
the previous sections we have expressed the correlation spectrum as a function
of detachment level and the underlying portfolio’s cumulative expected default
probability p rather than the term to maturity T . Given our assumption of the
static term structure of the hazard rates h these two formulations are equivalent.
In this section we prefer to emphasize the dependence on maturity horizon in
order to facilitate the comparison with base correlation models and also to
analyze the dependence on the level of hazard rates separately from the term
to maturity dimension.
In this framework, we can compare various dynamic and static loss generating models by comparing their correlation spectra, as well as the characteristic
dependencies of the correlation spectra on changes of model parameters. Of
course, the correlation spectrum of a static Gaussian copula model [19] is a flat
surface with constant correlation across both detachment level K and term to
maturity T . Any deviation from a flat surface is therefore an indication of a
non-trivial loss generating model, and we can judge which features of the model
are the important ones by examining how strong a deviation from flatness do
they lead to.
6.1
Models with static dependence structure
Let us begin with the analysis of one of the popular static loss generation models.
On Figure 8 we show the correlation spectrum computed for the Student-t
copula with linear correlation ρ = 0.3 and ν = 12 degrees of freedom. Student-t
copula is in the same elliptic family as the Gaussian copula but has non-zero
tail dependence governed by the degrees of freedom parameter. As a model of
single-period asset returns the Student-t distribution has been shown to provide
a significantly better fit to observations than the standard normal [20].
However, from the Figure 8 we can see that the static Student-t copula does
not generate a notable skew in the direction of detachment level K, and in fact
generates a mild downward sloping skew for very short terms, which is contrary
to what is observed in the market. The main reason for this is the rigid structure
29
0.5
0.5
Gaussian
Gaussian
Student-t(v =12)
Double-t(v =12, v =100)
m
0.45
Student-t(v =12)
Double-t(v =12, v =12)
i
m
Double-t(v =12, v =12
m
0.45
i
m
i
ρ
0.4
ρ
0.4
i
Double-t(v =12, v =100)
0.35
0.35
0.3
0.3
0.25
0
0.05
0.1
K
0.15
0.25
0.05
0.2
0.1
0.15
0.2
0.25
0.3
K
Figure 8: Correlation spectrum slices corresponding to 1-year (left chart) and 5-year
5
(right chart) horizons with default probabilities 0.02 and 1−(1 − 0.02) = 0.0961 for
Gaussian, Student-t Copula with 12 degrees of freedom, and Double-t Copulas with
(vm =12, vi =100) and (vm =12, vi =12) degrees of freedom. Linear correlation is
0.3 for all copulas.
of this model, with the tails of the idiosyncratic returns tied closely to the tails of
the market factor. This can be explicitly seen in the derivation of the Student-t
copula as a mixture model [27], and leads to a joint survival of all issuers when
the χ2 -distributed mixing variable takes on a large value (almost) regardless of
the realization of either the market factor or the idiosyncratic returns. Instead
of producing a varying degree of correlation depending on the default threshold,
the Student-t copula model simply produces a higher overall level of correlation.
On the other hand, the more flexible double-t copula model [17] produces
a significantly upward sloping skew, as can be seen from Figure 8. The main
feature of the double-t copula that is responsible for the skew is the cleaner
separation between the common factor and idiosyncratic returns — there is no
longer a single mixing variable which ties the two sources of risk together. The
presence of a fat-tailed market factor while the idiosyncratic returns can in fact
get efficiently diversified in the LHP framework naturally leads to an upward
sloping correlation skew. Recall that the higher values of the detachment level
K correspond to farther downside tails of the market factor where it dominates
the idisyncratic returns, resulting in a higher effective correlation.
Furthermore, by making the fully independent idiosyncratic returns more fat
tailed one achieves a steeper skew — compare the two examples of the doublet copula, with the degrees of freedom of the idiosyncratic returns set to 100
(i.e. nearly Gaussian case) and to 12 (i.e. strongly fat-tailed case), respectively.
Finally, we observe that the slope of the correlation skew gets flatter as the time
horizon grows. All of these features will have their close counterparts in the
dynamic models which we will consider next.
30
6.2
Multi-Period (Dynamic) Loss Generating Model
Let us now turn to loss generating models based on latent variables with multiperiod dynamics. We have concluded in the previous section that a clean separation of the market factor and the idiosyncratic returns appears to be a prerequisite for producing an upward sloping correlation skew. Fortunately, the
dynamic multi-variate models which we considered in section 3 all have this
property, both for single-period and for aggregated returns.
On Figure 9 we show the correlation spectrum computed for a loss generating model based on GARCH dynamics with Gaussian residuals, with a linear
correlation set to ρ = 0.3, and GARCH model parameters taken from the weekly
SP500 estimates in Appendix B.
As we can see, this model does exhibit a visible deviation from the flat correlation spectrum for short maturities. However, as we already noted in section 3,
the distribution of aggregate returns for the symmetric GARCH model quickly
converges to normal. Therefore, it is not surprising to see that the correlation
spectrum also flattens out fairly quickly and becomes virtually indistinguishable from a Gaussian copula for maturities beyond 5 years. Thus, we conclude
that the symmetric GARCH model with Gaussian residuals is inadequate for
description of liquid tranche markets where one routinely observes steep correlation skews at maturities as long as 7 and 10 years.
Based on the empirical results of 3.2 we know that a GARCH model with
Student-t residuals provides a better fit to historical time series of equity returns.
A natural question is whether allowing for such volatility dynamics can lead to a
persistent correlation skew commensurate with the levels observed in synthetic
CDO markets.
The results of section 3.2 suggest that the additional kurtosis of the singleperiod returns represented by the Student-t residuals does not matter very much
for aggregate return distributions at sufficiently long time horizons. Indeed,
Figure 10 shows that the GARCH model with Student-t residuals exhibits a
correlation skew that is quite a bit steeper at the short maturities, yet is almost
as flat and featureless at the long maturities as its non-fat-tailed counterpart —
there is a small amount of skew at 10 years, but it is too small compared to the
steepness observed in the liquid tranche markets. Thus, we conclude that one
has to focus on the dynamic features of the market factor process in order to
achieve the desired correlation skew effect.
Our next candidates are the TARCH models with either Gaussian or Studentt return innovations. We have seen in 3 that the asymmetric volatility dynamics
of these models leads to a much more persistent skewness and kurtosis of aggregated equity returns that actually grow rather than decay at very short horizons,
and survive for as long as 10 years for the range of parameters corresponding
to the post-1990 sample of SP500 weekly log-returns. Hence, our hypothesis is
that a latent variable model with TARCH market fynamics might be capable of
producing a non-trivial credit correlation skew for up to 10 year maturity.
31
T=1
T=5
T=7
T=10
0.55
0.6
0.5
0.5
0.45
ρ
0.4
ρ
0.3
0.4
0.35
0.2
0.3
0.3
0.2
K
10
8
0.1
0
2
4
0.25
6
0.2
T(y ears)
0
0
0.05
0.1
0.15
K
0.2
0.25
0.3
Figure 9: Correlation spectrum for GARCH model (α=0.045, β =0.948) with Gaussian shocks and the slices of the correlation spectrum for 1, 3, 5 and 7 year maturities.
T=1
T=5
T=7
T=10
0.55
0.6
0.5
0.5
0.45
ρ
0.4
ρ
0.3
0.4
0.35
0.2
0.3
0.3
10
0.2
K
6
0.1
0
2
0
8
4
0.25
0.2
T(y ears)
0
0.05
0.1
0.15
K
0.2
0.25
0.3
Figure 10: Correlation spectrum for GARCH model (α=0.045, β =0.948) with
Student-t shocks (v=8.3) and the slices of the correlation spectrum for 1, 3, 5 and
7 year maturities.
32
T=1
T=5
T=7
T=10
0.55
0.6
0.5
0.5
0.45
ρ
0.4
ρ
0.3
0.4
0.35
0.2
0.3
0.3
0.2
K
10
8
0.1
0
2
0
4
0.25
6
0.2
T(y ears)
0
0.05
0.1
0.15
K
0.2
0.25
0.3
Figure 11: Correlation spectrum for TARCH model (α=0.004, αD =0.094, β =0.927)
with Gaussian shocks and the slices of the correlation spectrum for 1, 3, 5 and 7 year
maturities.
The Figures 11 and 12 show the correlation spectra for the TARCH-based
loss generating models. The most immediate observation is that both versions of
the model produce a rather persistent correlation skew. Although the correlation
spectrum surface flattens out with growing term to maturity, the steepness
of the skew is still quite significant even at 10 years. Just as in the case of
the symmetric GARCH model, the fat-tailed residuals lead only to marginal
steepening of the correlation spectrum compared to the case with Gaussian
residuals.
Contrast these properties of the dynamic GARCH-based models with the
features of the static double-t copula. Upon a closer inspection of Figures 8
and 11 we can see that the TARCH model with Gaussian shocks and Gaussian idiosyncracies produces a slightly steeper 5-year correlation skew than the
double-t copula, even when the latter is taken with fat-tailed idiosyncracies.
When we turn on the Student-t return residuals for the market factor dynamics (see Figure 12) the differences in the 5-year skew become quite siginificant.
Thus, we conclude that the sources of persistent correlation skew are, in order
of their importance: 1) the independence of market factor and idiosyncratic
returns, 2) the asymmetry of market factor aggregate return distribution, 3)
fat tails in the market factor return dynamics, 4) fat tails in the idiosyncratic
return distribution.
6.3
Dependence on Model Parameters
As explained in the beginning of this section, we consider the correlation spectrum surface as an embodiment of the particular loss generating model. Each
such model contains various parameters some of which are empirically estimated
(e.g. the degrees of freedom of the Student-t distribution) and some of which are
calibrated to a particular problem at hand (e.g. the level of hazard rates and ex33
T=1
T=5
T=7
T=10
0.55
0.6
0.5
0.5
0.45
ρ
0.4
ρ
0.3
0.4
0.35
0.2
0.3
0.3
10
0.2
K
6
0.1
0
2
0
8
4
0.25
0.2
T(y ears)
0
0.05
0.1
0.15
K
0.2
0.25
0.3
Figure 12: Correlation spectrum for TARCH model (α=0.004, αD =0.094, β =0.927)
with Student-t shocks(v=8.3) and the slices of the correlation spectrum for 1, 3, 5 and
7 year maturities.
pected default probabilities for the collateral portfolio underlying the synthetic
CDO tranches under question). While the empirically estimated parameters
are not likely to change, the calibrated ones will do so quite frequently as the
market conditions change.
In particular, the implied hazard rates can and do change quite significantly
even for investment grade credit portfolios. Therefore, the analysis of the dependence of the correlation spectrum on the level of hazard rates has not only
an academic relevance as a matter of investigation of the model’s range of applicability, but also a practical importance due to reliance of many practitioners
on the base correlation methodology which normally takes the correlation skew
as an exogenous input and does not incorporate correlation skew adjustments as
the market spreads and implied hazard rates change. By contrast, the dynamic
multi-period models introduced in this paper produce the correlation spectrum
as an output of the model, and therefore can give a specific prediction regarding
the way the correlation skew is supposed to change when the model parameters
move.
As an example of such predictive behavior of the model consider the correlation spectrum dependence on the hazard rates depicted in Figure 13, where we
have shown a particular maturity slice, the 5-year skew, as a function of hazard
rates. From the visual comparison of Figures 13 and 11 it appears that the
dependence of a correlation skew for a fixed term to maturity but varying level
of hazard rates is very similar to the dependence of the correlation spectrum on
the term to maturity. The similarity is natural, as the first order effect is the
dependence on the level of the cumulative default probability which depends on
the product of h · T rather than on the hazard rate or the term to maturity
separately. For each level of this product, we get a specific level of the default
threshold in the latent variable credit risk model. The higher this threshold,
34
T=5 h=100bp
T=5 h=200bp
T=10 h=100bp
0.55
0.6
0.5
0.5
0.45
ρ
0.4
ρ
0.3
0.4
0.35
0.2
0.3
0.3
500
0.2
K
400
300
0.1
200
0
100
0.25
0.2
h(bp)
0
0.05
0.1
0.15
K
0.2
0.25
0.3
Figure 13: Left figure : the dependence of the 5-year correlation skew on the level
of the hazard rates. Right figure shows 3 correlation spectrum slices: 1) T=5-yr
h=100bp, 2) T=10-yr h=100bp, 3) T=5-yr h=200bp
the closer is the sampled region to the center of the latent variables distribution
and the less it is affected by the tail risk — thus leading to a lower level of the
credit correlation.
However, there is a second order effect which makes these two dependencies
somewhat different. It is related to the shape of the distribution of aggregate
returns for the market factor. Assuming that the parameters of the GARCH
process are the same in both cases, we can deduce that the dependence on the
term to maturity with fixed hazard rate should exhibit a faster flattening of
the correlation spectrum than the dependence on the hazard rate with fixed
term to maturity because the increasing aggregation horizon for market factor
returns leads to gradual convergence of its distribution towards normal and, as
a consequence, to progressively flatter correlation skew.
To make the visual comparison easier, we note that the effect of flattening
of the skew while going from a 5-year horizon to the 10-year horizon must be
compared against the flattening of the skew while going from 100bp hazard
rate to 200bp hazard rate. The right hand side Figure in 13 contains a direct
comparison of these three particular slices of the correlation spectrum surface
and confirms the intuition put forward above.
7
Summary and Conclusions
In this paper we have introduced and studied a new class of credit correlation
models defined via a dynamic portfolio loss generation process within a latent
variable approach where the latent variable follows a factor-ARCH with asymmetric TARCH volatility dynamics. We have shown this model to be superior
to alternative simpler characterizations of the time series processes including
symmetric GARCH volatility dynamics with Gaussian or Student-t residuals
35
when it comes to ability to produce a significant and persistent correlation skew
commensurate with the levels observed in the liquid synthetic CDO tranche
markets.
To build the foundation for our model, we have studied the time aggregation
properties of the multivariate dynamic models of equity returns. We showed that
the dynamics of equity return volatilities and correlations leads to significant
departures from the Gaussian distribution even for horizons measured in several
years. The asymmetry appears to "survive aggregation" longer than fat tails
do based on the parameters estimated from the real data. The main source of
skewness and kurtosis of the return distribution for long horizons is the dynamic
asymmetry of volatility response to return shocks or so-called leverage effect.
We introduced the notion of the correlation spectrum as a tool for comparing
the loss generating models, whether defined via a single-period (static) copula,
or via multi-period (dynamic) latent-variable framework, and for simple and
consistent approach to non-parametric pricing of CDO tranches. We showed
that for a portfolio loss distributions with smooth pdf the loss distribution can
be easily reconstructed from the correlation spectrum using its level and slope
along the K-dimention.
Importantly, in our dynamic loss generating model framework, the correlation spectrum is not only explained, but predicted — based on empirical parameters of the TARCH process and the parameters describing the reference
credit portfolio. The model also predicts a specific sensitivity of the correlation
spectrum to changes in various such parameters, including the hazard rate. The
structural inability of the static models to incorporate the changes in the base
correlation have been at the heart of the recent difficulties faced by these models
during the synthetic CDO market dislocation in April/May of 2005. While our
model is not likely to have given all the answers in such turbulent market conditions either, its ability to accommodate the changes in the correlation skew
could help the practitioners get a better handle on the fast moving markets.
One of the possible directions for generalization of our model is to move
from a single market factor to a multi-factor framework. The well-documented
importance of both macro and industry factors for explanation of equity returns
suggests that the same factors could be instrumental in getting a more accurate
model of credit correlations as well.
Whether in a single factor or a multi-factor setting, many of our conclusions
reflect the limitations of the large homogeneous portfolio approximation which
we have adopted in this paper. In particular, it is clear that even deterministic
but heterogenous idiosyncrasies, market factor loadings and hazard rates could
lead to significant changes in portfolio loss distribution and consequently to the
correlation spectrum of the model. An extension of our model to such heterogenous case is possible, although the computational efforts will increase very
significantly. Still, the promising features demonstrated by our approach even
in the LHP approximation suggest that despite the computational difficulties,
such extensions might be a worthy effort. In particular, the explicit modeling
of the heterogeneous reference portfolio would have been absolutely necessary
if one were to attempt to explain the tranche pricing during significant market
36
dislocations.
Another important simplification which we have made when discussing the
results of our model with regard to the correlation spectra is that we have only
considered the unconditional return distributions and have not explored the
effects of the initial shocks to either TARCH returns of volatility. From the
perspective of a credit investor this means that we have described the "equilibrium" (in a loose sense of that word) state of the tranche market, but not the
effects related to the relaxation towards the equilibrium. One could expect to
find interesting results in this line of research, which would be that much more
relevant given the credit market’s propensity to undergo unexpected short-term
dislocations as we have witnessed several times over the past couple of years.
Among the more practical questions that remain for future investigation are
the calculation of the deltas or hedge ratios of synthetic CDO tranches within our
framework, defining relative value measures for tranches reflecting the model’s
ability to produce the "fair" or "predicted" correlation spectrum.
References
[1] L. Anderson, J. Sidenius (2005): "Extensions to Gaussian Copula: Random
Recovery and Random Factor Loadings", Journal of Credit Risk, vol. 1, no.
1, p. 29
[2] International Swaps and Derivatives Association (2005): "ISDA Mid-Year
2005 Market Survey", press release published on September 28, 2005.
[3] Bollerslev, T., R.F. Engle, and D. Nelson (1994): "ARCH Models", in The
Handbook of Econometrics, vol. IV, eds. D.F. McFadden and R.F. Engle
III. North Holland.
[4] G. Gupton, C. Finger, M. Bhatia (1997): "CreditMetrics: Technical Document", RiskMetrics Group.
[5] Day, T. E., and C. M., Lewis (1992): "Stock Market Volatility and the
Informational Content of Stock Index Options." Journal of Econometrics
52, p. 267-287.
[6] Drost, F. C. and T. E. Nijman (1993): Temporal aggregation of GARCH
processes.Econometrica 61, p. 909—927.
[7] P. Embrechts, F. Lindskog and A. McNeil (2001): "Modeling dependence
with copulas and applications to risk management", Working paper, ETH,
Zurich
[8] Engle, R.F. (1982): Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom in‡ation. Econometrica 50, p.
987-1006.
37
[9] Engle, R. F. (2002): "New Frontiers for ARCH Models", Journal of Applied
Econometrics, 17, 425—446 (2002).
[10] Engle, R. F. (2002): "Dynamic Conditional Correlation — A Simple Class of
Multivariate GARCH Models", Journal of Business and Economic Statistics
[11] E. W. Frees and E. A. Valdez (1998): "Understanding Relationships Using
Copulas", North American Actuarial Journal, vol. 2, no. 1, p. 1
[12] R. Frey and A. McNeil, "Dependent defaults in models of portfolio credit
risk", Journal of Risk (forthcoming),
[13] Laurent, J.-P., & J. Gregory, 2003, Basket Default Swaps, CDOs and Factor Copulas, ISFA Actuarial School, University of Lyon, working paper
available on www.defaultrisk.com
[14] J. Gregory(editor), "Credit Derivatives: The Definitive Guide", Risk Books
in association with Application Networks, September 25, 2003
[15] Gordy, M. (2000): “A comparative anatomy of credit risk models,” Journal
of Banking and Finance, 24, 119—149.
[16] He, C., and T. Teräsvirta (1999a): “Properties of moments of a family of
GARCH processes,” Journal of Econometrics, 92, 173—192
[17] Hull, J., & A. White (2004): Valuation of a CDO and an nth to Default
CDS without Monte Carlo Simulation, Journal of Derivatives, 2, 8-23.
[18] D. O’Kane, M. Naldi, S. Ganapati, A. Berd, C. Pedersen, L. Schloegl,
R. Mashal, “Lehman Brothers Guide to Exotic Credit Derivatives” Risk,
special supplement, October 2003
[19] D. X. Li, "On Default Correlations: a Copula Function Approach", Journal
of Fixed Income, vol. (2000)
[20] Mashal, R., Naldi, M. and Zeevi, A. (2003) Extreme Events and Multiname Credit Derivatives. In Credit Derivatives The Definitive Guide, Risk
Waters Group, Great Britain, 313-338
[21] Greenberg A., Mashal, R., Naldi, M. and Schlogel, L. (2004) Tuning Correlation and Tail Risk to the Market Prices of Liquid Tranches. Quantitative
Credit Research Quarterly, Lehman Brothers, March 3-12.
[22] R. Merton, "On pricing of corporate debt: the risk structure of interest
rates", Journal of Finance, vol. 29 (1974) p. 449
[23] L. McGinty, E. Beinstein, R. Ahluwalia, M. Watts, "Credit Correlation: A
Guide", JPMorgan Credit Derivatives Strategy, March 2004
38
[24] Nelsen, R. B. 1999. An Introduction to Copulas. Vol. 139 of Lecture Notes
in Statistics, Springer, Berlin.
[25] P. J. Shonbucher, "Credit Derivatives Pricing Models: Models, Pricing and
Implementation", J. Wiley & Sons (2003)
[26] Schonbucher, P. J. and D. Schubert. 2001. Copula-dependent default risk
in intensity models. Working Paper, Department of Statistics, Bonn University.
[27] Dominic O’Kane and Lutz Schloegl (2003): "A Note on the Large Homogeneous Portfolio Approximation with the Student-t Copula", Finance and
Stochastics, Volume 9, No 4, p 577 - 584.
[28] Sklar, A. 1959. Fonctions de r epartition a n dimensions et leurs marges.
Publications de l‘Institut de Statistique de l‘Universit e de Paris, 8:229-231.
[29] O. Vasicek, "Loan Portfolio Value", Risk Magazine, December 2002, pp
160-162
[30] O. Vasicek, "A Series Expansion for the Bivariate Normal Integral", working paper(1998), KMV
[31] Zakoïan, J.-M. (1994). "Threshold Heteroskedastic Models". Journal of
Economic Dynamics and Control 18, 931-955.
39
A
Kurtosis and Skewness of Aggregated TARCH
Returns
In this notes we analyze kurtosis and skewness of aggregated returns RT =
T
X
√1
rt when rt is assumed to follow TARCH(1,1) process
T
t=1
rt = σt εt
2
2
2
σt2 = (1 − ζ) σ 2 + αrt−1
+ αD rt−1
1{rt−1 ≤0} + βσt−1
where returns innovations εt are assumed to be iid, have zero mean and unit variance. We are interested in variance, skewness and kurtosis of time aggregated
returns. To make sure that those moments are finite we need corresponding
moments of the return innovations to be finite. Particulaly, we assume that εt
has finite kurtosis. Let us introduce the following notations for the central and
truncated moments of εt
mε ≡ E (εt ) = 0
¡ ¢
vε ≡ E ε2t = 1
¡
¢
vεd ≡ E ε2t 1{εt ≤0}
¡ ¢
sε ≡ E ε3t
¡
¢
sdε ≡ E ε3t 1{εt ≤0}
¡ ¢
kε ≡ E ε4t
¡
¢
kεd ≡ E ε4t 1{εt ≤0}
Lemma 7 The following recursions hold for TARCH(1,1) model
¡
¢
¡
¢
2
2
covt−1 rtk , rt+u
= ρcovt−1 rtk , rt+u−1
for u>1
¡ k 2 ¢
¡ k+2 ¢
¡
¢
+ αD vart−1 rtk+2 1{rt ≤0}
covt−1 rt rt+1 = αvart−1 rt
Proof.
¡
¢
¡ £
¤¢
2
2
2
2
covt−1 rtk , rt+u
+ αD rt+u−1
1{rt+u−1 ≤0} + βσt+u−1
= covt−1 rtk (1 − ζ) σ2 + αrt+u−1
¡
¢
¡
¢
¡
¢
2
2
2
1{rt+u−1 ≤0} + βcovt−1 rtk , σt+u−1
+ αD covt−1 rtk , rt+u−1
= 0 + αcovt−1 rtk , rt+u−1
if u>1 then
If u=1 then
¡
¢
¡
¢
2
2
covt−1 rtk , rt+u−1
1{rt+u−1 ≤0} = vεd covt−1 rtk , rt+u−1
¡
¢
¡
¢
2
2
= covt−1 rtk , rt+u−1
covt−1 rtk , σt+u−1
¡
¢
2
covt−1 rtk , σt+u−1
=0
40
Proposition 8 Suppose 0 ≤ ζ < 1 and the return innovations have finite skewness, sε , and finite "truncated" third moment, sdε , then conditional third moment of T-period aggregate return Rt,t+T has the following representation for
TARCH(1,1)
3
Et Rt,t+T
=
1
T
s
3/2 ε
T
X
u=1
¡ 3 ¢
Et σt+u
+
3
T 3/2
T
¡
¢X
1 − ζ T −u ¡ 3 ¢
αsε + αD sdε
Et σt+u
1−ζ
u=1
In addition if Eσt3 is finite then unconditional skewness of Rt,t+T is given by
∙
¸ ³ ´
3
T
¢
ERt,t+T
1
1 ¡
σt 3
d T (1 − ζ) − 1 + ρ
=
s
+
3
+
α
s
E
αs
ST ≡
ε
ε
D
ε
2
(1 − ζ)2
σ
E(Rt,t+T
)3/2
T 1/2
T 3/2
Proof. Using Lemma 7 we have
⎛
à t+T
!3
X
X
ru
= Et ⎝
Et
u=t+1
t+1≤t1 ≤t2 ≤t3 ≤t+T
=
T
X
u=1
=
T
X
T
X
rt1 rt2 rt3 ⎠
t+1≤t1 <t2 ≤t+T
¡ 3 ¢
Et rt+u
+3
u=1
=
X
3
Et rt+u
+
⎞
X
¡
¢
3Et rt1 rt22
t+1≤t1 <t2 ≤t+T
³
³
¡ ¢
ζ t2 −t1 −1 αEt rt31 + αD Et rt31 1{rt
1 ≤0
}
´´
T
X
¡ 3 ¢
¡ 3 ¢
¡ 3
¢¢
1 − ζ T −u ¡
Et rt+u
1{rt+u ≤0}
+3
+ αD Et rt+u
αEt rt+u
1−ζ
u=1
u=1
= sε
T
¡ 3 ¢ ¡
¢X
1 − ζ T −u ¡ 3 ¢
Et σt+u
+ αsε + αD sdε
Et σt+u
1−ζ
u=1
u=1
T
X
Using the law of iterated expectations
⎛ Ã
à t+T
!3
!3 ⎞ ∙
¸
t+T
X
X
¢ T (1 − ζ) − 1 + ζ T
¡
E
E (σt )3
ru
= E ⎝Et
ru ⎠ = T sε + 3 αsε + αD sdε
2
(1
−
ζ)
u=t+1
u=t+1
ST is then computed using the simple formula for the unconditional variance
2
E(Rt,t+T
) = σ2 .
To derive unconditional kurtosis we define the following unconditional autocorrelations
2
, rt2 )
γn = γ−n = corr(rt−n
ϕn = corr(rt−n , rt2 ) for n ≥ 1
¡
¢
ψi,j ≡ E rt−i rt−j rt2 for 1 ≤ j < i
41
Lemma 9 γn , ϕn and ψi,j decay exponentially as n and i − j increase
γn = ζγn−1 = ζ n−1 γ1 for n ≥ 1
ϕn = ζϕn−1 = ζ n−1 ϕ1 for n ≥ 1
ψi,j = ζψi−1,j−1 = ζ j−1 ψi−j+1,1 for 1 ≤ j < i
where γ1 , ϕ1 and ψk,1 are given by
¡
¢
γ1 = α (kr − 1) + αD krd − vrd + βkr /kε
with
krd =
vεd
=
ϕ1 = αsr + αD sdr
¡
¡
¢
¢
ψk,1 = αE rt−k+1 rt3 + αD E rt−k+1 rt3 1{rt ≤0}
E (rt2 1{rt ≤0} )
,
Ert2
rt4 1{rt ≤0}
(Ert2 )2
E(
)
sr =
E (rt3 )
(Ert2 )3/2
,
sdr
=
³
E rt3 1{r
T <0}
(Ert2 )3/2
´
, kr =
E (rt4 )
(Ert2 )2
and
.
Proposition 10 If
¡
¢
ζ ≡ E β + αε2t + αD ε2t 1{εt ≤0} = β + α + αD vεd < 1
¢2
¡
ξ ≡ E β + αε2t + αD ε2t 1{εt ≤0} = β 2 + α2 kε + α2D kεd + 2αβ + 2αD βvεd + 2ααD kεd < 1
then unconditional kurtosis of rt , K1 , is finite and
K1 ≡
Ert4
2
(Ert2 )
= kε
1 − ζ2
1−ξ
Proof. If the 4th moment of rt exists then the following equation must hold
¡ ¢ ¡ ¢
Ert4 = E ε4t E σt4
¡
¢2
2
2
2
+ αD rt−1
1{rt−1 ≤0} + βσt−1
= kε E (1 − ζ) σ 2 + αrt−1
¡ 2
¢
2
2
2
= kε ((1 − ζ) σ 4 + 2 (1 − ζ) σ 2 E αrt−1
+ αD rt−1
1{rt−1 ≤0} + βσt−1
¡ 2
¢2
2
2
+ αrt−1
+ αD rt−1
1{rt−1 ≤0} + βσt−1
)
³
´
2
4
= kε (1 − ζ) σ4 + 2 (1 − ζ) ρσ 4 + ξEσt−1
Therefore Ert4 nessecerily solves
¡
¢
Ert4 = kε 1 − ζ 2 σ 4 + ξErt4
Proposition 11 If the distibution of εt is symmetric and αD = 0 then unconditional kurtosis of RT , if exists, is given by the following formula:
1
γ1 T (1 − ζ) − 1 + ζ T
for T > 1
(K1 − 3) + 6 2
T
T
(1 − ζ)2
1 − ζ2
K1 = kε
1−ξ
KT = 3 +
42
(66)
(67)
where kε is unconditional kurtosis of εt and
¡
¢2
2 d
ξ ≡ E β + αε2t + αD ε2t 1{εt ≤0} = β 2 + α2 kε + αD
kε + 2αβ + 2αD βvεd + 2ααD kεd .
¡ 2
¢
¡
¢
γ1 ≡ corr rt−1 , rt2 = α (kr − 1) + αD krd − vrd + βkr /kε
Proof.
E
Ã
t+T
X
ru
u=t+1
!4
=
=
T
X
¡ 4 ¢
+6
E rt+u
X
u=1
t+1≤t1 <t2 ≤t+T
T
X
X
¡ 4 ¢
E rt+u
+6
u=1
t+1≤t1 <t2 ≤t+T
¢
¡
E rt21 rt22
£
¡
¢
¡ ¢ ¡ ¢¤
cov rt21 , rt22 + E rt21 E rt22
¡ ¢
¡ 2
¢
T (T − 1) ¡ 2 ¢2
= T E rt4 + 6
, rt2
E rt + 6cov rt−1
2
X
ζ t2 −t1 −1
t+1≤t1 <t2 ≤t+T
¡ ¢
¡ 2
¢ T (1 − ζ) − 1 + ζ T
T (T − 1) ¡ 2 ¢2
= T E rt4 + 6
, rt2
E rt + 6cov rt−1
2
(1 − ζ)2
substituting the derived 4th moment into the definition of the kurtosis KT =
!4
à t+T
X
¡ ¢2
ru /E rt2 completes the proof.
E
u=t+1
43
B
Estimation Results for SP500
Table 1 SP500 moments.
Daily
sdr
kr
vrd
sr
Sample period
sr
1962-2004
1990-2004
-1.40
-0.11
-2.43
-1.14
39.83
6.67
0.53
0.51
-0.55
-0.64
Weekly
sdr
kr
-1.35
-1.36
7.01
6.10
vrd
0.55
0.56
SP500 moments(After trimming 0.1% of extreme positive and negative returns )
1962-2004
0.05
-1.03
5.95
0.50 -0.39 -1.18 5.26 0.54
1990-2004
0.04
-1.01
5.56
0.50 -0.64 -1.36 6.10 0.56
Table 2
Estimated parameters of GARCH(1,1)/TARCH(1,1) with Gaussian/Student-T shocks
on daily(D) and weekly(W) SP500 returns for.[01/01/1990-12/31/2004].
α
αD
β
ν
α
αD
β
ν
D
W
D
W
D
W
D
W
0.056
0.044
0.007
0.112
0.918
0.047
0.045
0.006
0.095
0.941
8.21
0.004
0.094
0.927
8.31
-
-
0.941
0.953
0.007
0.100
0.933
-
-
-
After trimming 0.1%
0.062 0.094 0.023
0.067
0.935 0.896 0.941
-
-
-
-
0.951
7.25
0.948
7.77
of extreme positive and
0.032 0.062 0.088
0.112
0.895 0.936
0.90
8.82
10.64
negative returns
0.022 0.030
0.073 0.104
0.938
0.90
9.63
11
Table 3
Estimated parameters of GARCH(1,1)/TARCH(1,1) with Gaussian/Student-T shocks
on daily(D) and weekly(W) SP500 returns for.[01/01/1962-12/31/2004].
α
αD
β
ν
α
αD
β
ν
D
W
D
W
D
W
D
W
0.076
0.107
0.037
0.136
0.877
0.064
0.09
0.027
0.072
0.934
8.44
0.032
0.106
0.894
10.19
-
-
0.923
0.886
0.029
0.081
0.928
-
-
-
After trimming 0.1%
0.051 0.044 0.004
0.093
0.945 0.953 0.940
-
-
-
-
0.934
7.86
0.897
9.26
of extreme positive and
0.007 0.046 0.045
0.112
0.918 0.952 0.947
7.78
7.77
44
negative returns
0.004 0.004
0.096 0.094
0.941 0.926
9.011
8.31
C
Monte Carlo Simulation of Portfolio Loss under LHP Assumption
Because of the one factor structure of the model we can use LHP setup described
in proposition 1 to calibrate the loss of a large homogeneous portfolio using the
distribution of the aggregated market return generated by TARCH(1,1) model.
The latent variables are assumed to have symmetric one factor structure with
the factor following TARCH(1,1) model. We calibrate the loss of the portfolio
using the one factor GARCH model described in and the formula 35 for LHP
loss
µ
¶
³
_´
dT − bRm,T
√
LT = 1 − R Φ
1 − b2
where
• Rm,T =
√1
T
T
X
rm,u is return over horizon T generated using aggregation
u=1
of simulated TARCH(1,1) returns with unconditional volatility equal to 1
√
• dT is calibrated so that the probability of Ri,T = bRm,T + 1 − b2 ET
hitting dT is equal to single name default probability pT
³
´
p
P bRm,T + 1 − b2 ET ≤ dT = pT
• b is the factor loading that is chosen to match a given unconditional linear
correlation ρ = b2
To calculate the expected tranche losses generated by the model and to
calibrate dT we use I = 100, 000 independent Monte Carlo simulations of the
factor and then use corresponding sample moments:
Ã
!
(i)
I
dT − bRm,T
1X
√
dT solves
Φ
= pT
I i=1
1 − b2
Ã
Ã
!!
(i)
I
³
_´
dT − bRm,T
1X
√
EL(0,K] =
f(0,K]
1−R Φ
I i=1
1 − b2
45